Question: 2018 Fall Final - 14
Let $n$ and $k$ be integers with $1 \leq k \leq n$.
You are given two bitstrings $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ of length $n$. In both bitstrings, each bit is 0 with probability 1/2, and 1 with probability 1/2 (independent of all other bits).
Consider the bitstring $$ a_1 \cdot b_1,a_2 \cdot b_2,\dots,a_n \cdot b_n $$ where $a_i \cdot b_i$ is the result of multiplying the two bits $a_i$ and $b_i$. What is the probability that this bitstring contains exactly $k$ many 1's?
You are given two bitstrings $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ of length $n$. In both bitstrings, each bit is 0 with probability 1/2, and 1 with probability 1/2 (independent of all other bits).
Consider the bitstring $$ a_1 \cdot b_1,a_2 \cdot b_2,\dots,a_n \cdot b_n $$ where $a_i \cdot b_i$ is the result of multiplying the two bits $a_i$ and $b_i$. What is the probability that this bitstring contains exactly $k$ many 1's?
(a)
${n \choose k} \cdot \left. 3^{n - k} \middle/ 4^n \right.$
(b)
None of the above.
(c)
${n \choose k} \cdot \left. 4^{n - k} \middle/ 3^n \right.$
(d)
${n \choose k} \cdot \left. 4^{n - k} \middle/ 3^k \right.$